ENDERSGRANGER Procedure

@EndersGranger performs the Enders-Granger test for threshold unit root behavior from Enders and Granger(1998). A related procedure for cointegration is @EndersSiklos.

This uses one of three "attractor" functions:

\(A(t) = \left\{ {\begin{array}{*{20}{c}} 0 \\ {{a_0}} \\ {{a_0} + {a_1}t} \\ \end{array}} \right.\)

controlled by the ATTRACTOR option. The regression run is

\(\Delta {y_t} = {\rho _1}{D_t}\left( {{y_{t - 1}} - A\left( {t - 1} \right)} \right) + {\rho _2}(1 - {D_t})\left( {{y_{t - 1}} - A(t - 1)} \right) + {\rm{lags}}\,{\rm{of}}\,\Delta y\)

where \({D_t}\) is the dummy for when the threshold series \(< \tau\). There are two options for the threshold series: either \({{y_{t - 1}}}\) or \(\Delta {y_{t - 1}}\), where the latter is called "Momentum TAR". \({\rho _1} = {\rho _2} = 0\) implies unit root behavior. \({\rho _1}\) and \({\rho _2}\) negative means that the process is "driven" towards the attractor. If they're negative but equal, there is no threshold effect. The procedure calculates three test statistics: the maximal t-stat on \({\rho}\), a joint test for zero, and a test for equality.

@EndersGranger( options ) y start end

Parameters

y	series to analyze
start, end	range of y to use. By default, the defined range of y.

Options

LAGS=# of lags on the differences [1]

ATTRACTOR=ZERO/[CONSTANT]/TREND

MODEL=[TAR]/MTAR

With MODEL=TAR, the threshold series is the lag of y. With MODEL=MTAR, it's the lagged difference of y.

TAU=threshold value [not used, chosen by search]

PI=fraction of high and low empirical values for the threshold series which are omitted in the search [.15]

If you provide a value of TAU, the regressions are done with that fixed value for tau. If you don't, the values of the threshold series (excluding the PI fraction at each end) are searched for the one that minimizes the sum of squared residuals.

[PRINT]/NOPRINT

TITLE=title for test report ["Enders-Granger Test"]

Variables Defined

%%BREAKVALUE	threshold value chosen or input (REAL)
%%EQUALSTAT	test for equality for the \(\rho\)'s (REAL)
%%PHISTAT	joint test for zero on the \(\rho\)'s (REAL)
%%TMAX	maximum t-statistic on the \(\rho\)'s (REAL)

Example

* Enders, Applied Econometric Time Series, 4th edition

* Example from Section 7.11, pp 464-465

* Threshold unit root test

open data granger.xls

cal(q) 1958:1

data(format=xls,org=columns) 1958:01 1994:01 date r_short r_10

set spread = r_10-r_short

set ds = spread-spread{1}

@dfunit(maxlags=4,method=aic) spread

linreg ds

# constant spread{1} ds{1}

@EndersGranger(lags=1,attractor=constant) spread

@EndersGranger(lags=1,attractor=constant,model=mtar) spread

Sample Output

Enders-Granger Test

TAR Model

Lags 1

Attractor -0.2700

T-Max -1.5904

Phi 7.7912

Equality 6.4230

Coeff Std Error

Above -0.0656 0.0412

Below -0.2858 0.0780

DY{1} 0.1717 0.0829

Enders-Granger Test

M-TAR Model

Lags 1

Attractor 1.6400

T-Max -0.1445

Phi 11.4462

Equality 12.2402

Coeff Std Error

Above -0.2994 0.0630

Below -0.0071 0.0495

DY{1} 0.0161 0.0879